Uniqueness of approximate solutions of Beltrami equations (Q2901733)

The present paper is devoted to the study of the Beltrami equation with degeneration. Let \(D \) be a domain in \({\mathbb C}\). A mapping \(f:D\rightarrow {\mathbb C}\) is said to be an approximate solution of the Beltrami equation \(f_{\overline z}=\mu(z)\cdot f_z\), if there exists a sequence \(f_n\) of solutions of the same Beltrami equation with complex characteristics \(\mu_n(z)\) such that \(\mu_n(z)=\mu(z)\) when \(| \mu(z)| \leq 1-\frac 1n\) and \(\mu_n(z)=0\) otherwise. A solution \(f\) of the Beltrami equation is said to be regular if it satisfies the above equality almost everywhere, if \(f\in W_{\text{loc}}^{1,1}\) and its Jacobian is non-degenerate a.e. Theorem 4.1 states that every approximate solution of the Beltrami equation is a regular solution. Theorem 5.1 presents a factorization of every approximate solution \(f\) as \(f=\varphi\circ g\) with a suitable function \(g\) which is also a solution of the Beltrami equation and a suitable conformal mapping \(\varphi\), provided that the corresponding maximal dilatation \(K_{\mu}(z)\) has a locally integrable majorant \(Q(z)\). Corollary 6.1 states the existence of the unique approximate solution of the Beltrami equation in a domain \(D\subset {\mathbb C}\) provided that \(| \mu(z)| <1\) a.e., \(K_{\mu}(z)\) is locally integrable in \(D\) and the so-called tangential dilatation \(K_{\mu}^T(z, z_0)\) has a majorant \(Q_{z_0}(z)\) in the class of finite mean oscillation in some neighborhood \(U_{z_0}\) of any point \(z_0\in D\). As usual, uniqueness of the solution is understood up to pre-composition with a conformal mapping. Finally, two conjectures relating approximate solutions and some other special solutions are proved.